We present the star formation rates (SFRs) of a sample of 109 galaxies with X-ray selected active galactic nuclei (AGN) with moderate to high X-ray luminosities (L(2-8keV)= 10^42-10^45 erg/s), at redshifts 1 < z < 4.7, that were selected to be faint or undetected in the Herschel bands. We combine our deep ALMA continuum observations with deblended 8-500{\mu}m photometry from Spitzer and Herschel, and use infrared (IR) SED fitting and AGN – star formation decomposition methods. The addition of the ALMA photometry results in an order of magnitude more X-ray AGN in our sample with a measured SFR (now 37%). The remaining 63% of the sources have SFR upper limits that are typically a factor of 2-10 times lower than the pre-ALMA constraints. With the improved constraints on the IR SEDs, we can now identify a mid-IR (MIR) AGN component in 50% of our sample, compared to only ~1% previously. We further explore the F870{\mu}m/F24{\mu}m-redshift plane as a tool for the identification of MIR emitting AGN, for three different samples representing AGN dominated, star formation dominated, and composite sources. We demonstrate that the F870{\mu}m/F24{\mu}m-redshift plane can successfully split between AGN and star formation dominated sources, and can be used as an AGN identification method.